
\section[Category....]{Categories...}
\begin{definition}[Section and retact]\label{Section}
Let $\bcC$ be a category, $p:X'\rightarrow X$ in $\bcC$, we say $s:X\rightarrow X' $ is a section of $p$, and that $p$ is a retract of $s$, if $p\circ s=id_X$.
\end{definition}
%\noindent Notice that a morphism $f$ might have more than one section, look for the case arisen from sections of vector bundles of varieties.
\subsection{Canonical Topology}
%\gray

\begin{definition}[Effective epimorphism]\label{Effective epimorphism}
Let $\bcC$ be a category with all pullbacks, $f:V\rightarrow U$ in $\bcC$, we say that $f$ is an effective epimorphism if the follwoing diagrams are exact, $\forall W\in \bcC$:
$$
\xymatrix{ Hom_{\bcC}(U,W)\stackrel{f^{\ast}}{\longrightarrow} Hom_{\bcC}(V,W)\ar@<-3pt>[rr]_!{"1,1";"1,2"}{\pi^{\ast}_2}\ar@<3pt>[rr]^!{"1,1";"1,2"}{\pi^{\ast}_1}& &Hom_{\bcC}(V\times_U V,W)}
$$
Where $(\pi_1,\pi_2)$ is the kernel pair of $f$.
Let $\bcU=\{f_i:U_i\rightarrow U|i\in I\}$ be a family of morphisms of $\bcC$, we say that $\bcU$ is a family of effective epimorphisms if the following diagrams are exact, $\forall W\in \bcC$:
\begin{equation}\label{FUEEDiagram}
\xymatrix{ Hom_{\bcC}(U,W)\stackrel{\displaystyle\prod_{i\in I}f_i^{\ast}}{\longrightarrow} \displaystyle\prod_{i\in I}Hom_{\bcC}(U_i,W)\ar@<-3pt>[rr]_!{"1,1";"1,2"}{\displaystyle\prod_{i,j\in I}\pi_{2,i,j}^{\ast}}\ar@<3pt>[rr]^!{"1,1";"1,2"}{\displaystyle\prod_{i,j\in I}\pi_{1,i,j}^{\ast}}& &\displaystyle\prod_{i,j\in I}Hom_{\bcC}(V\times_U V,W)}
\end{equation}
\end{definition}
For $f$, since the diagram is exact then $Hom_{\bcC}(U,W)\stackrel{f^{\ast}}{\rightarrow} Hom_{\bcC}(V,W)$ is an injection, i.e. $f$ is an epimorphism. Same argument implies that $\bcU$ is a family of epimorphisms. Same argument applies for the family $\bcU$.
\begin{definition}[Universal Effective epimorphism]\label{Universal Effective epimorphism}
Let $\bcC$ be a category with all pullbacks, $\bcU=\{f_i:U_i\rightarrow U|i\in I\}$ be a family of morphisms of $\bcC$, we say that $\bcU$ is a family of universal effective epimorphisms if $\bcU_f:=\{\pi_V:U_i\times_U V\rightarrow V|i\in I\}$ is a family of effective epimorhpisms for each $f:V\rightarrow U$.
\end{definition}
\begin{lemma}
Let $\bcC$ be a category with all pullbacks, $\bcU=\{f_i:U_i\rightarrow U|i\in I\}$ a family of universal effective epimorphisms, then $\bcU$ is a family of effective morphisms.
\end{lemma}
\begin{proof}
Put $U=V$ and $f=id_U$, then we have the pullback diagram
$$
\xymatrix{U_i\ar@{-->}[r]^{f_i}\ar@{-->}[d]_{id_{U_i}}&U\ar[d]^{id_U}\\
U_i\ar[r]_{f_i}&U}
$$
Hence, $\bcU$ is a family of effective morphisms.
\end{proof}
\begin{lemma}\label{UEEComposition}
Let $\bcC$ a category with pullbacks, then the composition of epimorphisms/EE/UEE is epimorphisms/EE/UEE.
\end{lemma}
\black
\subsection{Limits}
\begin{lemma}
Let $\bcC$ a category with fibre products, then $(U\times_X V)\times_U(U\times_X W)\cong ((U\times_X V)\times_X W)\cong (U\times_X (V\times_X W))$.
\end{lemma}
\begin{proof}
\tcr{type}
\end{proof}
